Star-Comb Lemma

Question

I cannot understand that how can we apply Zorn's lemma here. What is the order set?

Answer 1

The partially ordered set is the set of trees such that every edge lies on a path in that tree between two elements of $U$. The ordering is by inclusion. It satisfies the upper bound property because if we have a chain of such trees, we can take the union of their edges. This is a tree because any loop is a finite collection of edges and those edges will all be in some member of the chain. It is also clear that any edge lies on a path between two elements of $U$ ( that path being contained in the union ). Therefore the union is an element of the partially ordered set and is clearly greater than ( ie. contains ) every element of the chain. so the partially ordered set satisfies the conditions for Zorn's Lemma.